2.1 Language
description
Transcript of 2.1 Language
Basic definitions
(1) alphabet a finite set of symbols. ex) T1 = { ㄱ , ㄴ , ㄷ ,..., ㅎ , ㅏ , ㅑ , … , ㅡ , ㅣ }
T2 = {A,B,C, … ,Z} T3 = {auto, break, case, … , while}
(2) string(or sentence, word) a sequence of symbols from some alphabet T.
(3) length the number of symbols in the string. denoted by |ω|
2.1 Language
꼭 기억해야 할 세 가지 개념
1. 언어의 정의
2. 문법의 정의 및 개념
3. 인식기의 의미
(4) empty string a string consisting of no symbols. denoted by ε or λ.
(5) T* denotes the set of all strings of symbols over the alphabet T, including the empty string.
T+ = T* - {ε}
T* : T star
T+ : T dagger (6) Language is any set of strings over an alphabet.(Text
p.40) (or A Language L over the alphabet T is a subset of T*
.)
L T⊆ *
More definitions
(1) concatenation
u = a1a2a3...an, v = b1b2b3...bm , u • v = a1a2a3...anb1b2b3...bm
u • v 를 보통 uv 로 표기 . uε= u = εu ∀u,v T∈ *, uv T∈ *. |uv| = |u| + |v|
(2) an represents n a's. a0 = ε
(3) the reversal of a string ω, denoted ωR is the string ω written in reverse order:
i.e., if ω = a1a2...an then ωR = anan-1...a1.
Text p. 44
(ωR)R=ω
(4) language product
LL' = {xy| x L and y L'}∈ ∈
(5) The powers of a language L are defined recursively by:
L0 = {ε}
Ln = LLn-1 for n 1.
(6) L* : reflexive transitive closure
= L0 L∪ 1 L∪ 2 ... L∪ ∪ n ∪… =
(7) L+ : transitive closure
= L1 L∪ 2 ∪... L∪ n ...∪
= L* - L0
Language 문장 (sentence) 들을 원소로 갖는 집합 언어를 어떻게 표현할 것인가 ?
Grammar terminal : 정의된 언어의 알파벳 nonterminal :
스트링을 생성하는 데 사용되는 중간 과정의 심볼 언어의 구조를 정의하는데 사용
grammar symbol (V)
2.2 Grammar
Text p. 46 G = (VN, VT, P, S)
VN : a finite set of nonterminal symbols
VT : a finite set of terminal symbols
VN ∩ VT = , VN V∪ T = V
P : a finite set of production rules α → β, α V∈ +, β V∈ * lhs rhs
S : start symbol(sentence symbol)
[ 예 ] G = ( {S, A}, {a, b}, P, S ) Text p.47 [ 예제 2.8]
P : S → aAS S → a
A → SbA A → ba A → SS
⇒ S → aAS | a
A → SbA | ba | SS
Derivation
1. ⇒ : "directly produce" or "directly derive"
if α → β P and ∈ , δ V∈ * then
αδ ⇒ βδ
2. ⇒ : Suppose α1,α2,...,αn V∈ * and α1 ⇒α2 ⇒ … ⇒αn,
then α1 ⇒ αn
(zero or more derivations)
3. ⇒ : one or more derivations.
cf) → : production rule 에서 사용 .
“may be replaced by”
⇒ : derivation 할 때 사용한다 .
Inroduction to FL theory
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+
*
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L(G) : Language generated by grammar G
L(G) = {ω | S ⇒ ω, ω V∈ T*}
☞ ω is a sentential form of G if S ⇒ ω and ω V∈ *.
ω is a sentence of G if S ⇒ ω and ω V∈ T*.
P : S → aA | bB | ε
A → bS
B → aS
S ⇒ abba 유도 과정
S ⇒ aA ( 생성규칙 S → aA) ⇒ abS ( 생성규칙 A → bS) ⇒ abbB ( 생성규칙 S → bB) ⇒ abbaS ( 생성규칙 B → aS) ⇒ abba ( 생성규칙 S → )
S ⇒ aA ( 생성규칙 S → aA) ⇒ abS ( 생성규칙 A → bS) ⇒ abbB ( 생성규칙 S → bB) ⇒ abbaS ( 생성규칙 B → aS) ⇒ abba ( 생성규칙 S → )
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G1 = ( {S}, {a}, P, S ) 을 이용하여 L(G1)
P : S → a | aS
L (G1) = { an | n 1 }
Language design
Grammar Language
generation
design
Text p. 46
G = ( {A, B, C}, {a, b, c}, P, A)
P : A → abc A → aBbc
Bb → bB Bc → Cbcc
bC → Cb aC → aaB
aC → aa
L(G) = { anbncn | n 1 }
(===>) ex1) S → 0S1 | 01
ex2) S → aSb | c
ex3) A → aB
B → bB | b
ex4) A → abc A → aBbc Bb → bB Bc → Cbcc
bC → Cb aC → aaB
aC → aa
Grammar Design L = { an | n 0 } 일 때 문법 :
A → aA | ε L = { an | n 1 } 일 때 문법 :
A → aA | a Embedded production
A → aAb
ex1) L1 = { anbn | n 0 }
ex2) L2 = { 0i1j | i j, i,j 1 }
ex3) Constructs of Conventional PL
1) 파스칼 언어의 상수 정의 부분 :
상수정의 부분은 CONST 라는 예약어로 시작하며 하나의 상수
정의는 a=b 의 형태를 갖는다 . 여기서 , a 는 identifier 를 b 는
상 수 를 나 타 내 는 terminal 심 벌 이 다 . 상 수 정 의 부 분 은
선택적이며 각각의 상수정의는 ; 으로 구분한다 .
다음은 상수정의 부분의 예이다 .
CONST ON = TRUE;
OFF = FALSE;
EPSILON = 1.0E-10;
2) C 언어의 정수 선언 부분 :
정수선언 부분은 여러 개의 정수선언으로 구성되며 하나의 선언은 int a,a,a; 와 같은 형태를 갖는다 . 여기서 a 는 임의의 identifier 를 나타낸다 .
그리고 ; 으로 각각의 선언을 구분한다 . 예를 들어 , int i,j; int
sum; 과 같다 .
※ 문법을 고안할 때 , nonterminal 의 이름은 구문 구조를 대변할 수 있는 명칭으로 쓰는 것이 바람직하다 .
In order to prove that a grammar generates a language L
i) Every sentence generated by the grammar is in L. ii) Every string in L can be generated by the grammar.
교과서 55 쪽[ 예제 2.16]
proof) (=>) Every sentence derivable from S is balanced. (<=) Every balanced string is derivable from S.
G = ( { S }, { ( , ) }, {S → (S)S |ε}, S )
⇔ All strings of balanced parentheses.
(=>) Every sentence derivable from S is balanced. (i.e., S ⇒ ω, ω: balanced) By induction on the number of steps in a derivation.
i) n = 1 일 때 , S ⇒ ε, ε is surely balanced.
ii) Suppose that all derivations of fewer than n steps produce balanced sentences.
iii) Consider a leftmost derivation of exactly n steps. S ⇒ (S)S ⇒ (x)S ⇒ (x)y By the hypothesis x, y : balanced. Thus (x)y balanced.
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(<=) Every balanced string is derivable from S.
By induction on the length of a string.
i) |ω| = 0, S ε⇒
(the empty string is derivable from S.)
ii) Assume that every balanced string of length less than 2n is derived from S.
iii) Consider a balanced string ω of length 2n.
Let (x) : shortest prefix of ω being balanced.
Thus ω = (x)y, where x, y : balanced.
Since |x|, |y | < 2n, they are derivable from S by inductive hypothesis.
Thus S (S)S (x)S (x)y = ω⇒ ⇒ ⇒
Therefore, (x)y is also derivable from S.* *
Noam Chomsky According to the form of the productions. α → β P∈
Type 0 : No restrictions(unrestricted grammar) Type 1 : Context-sensitive grammar(CSG).
→ β, | | | β| Type 2 : Context-free grammar(CFG).
A → , where A : nonterminal, V∈ *. Type 3 : Regular grammar(RG).
A → tB or A → t, (right-linear)
A → Bt or A → t, (left-linear)
where, A, B : nonterminal, t V∈ T*.
2.3 Chomsky Hierarchy
REL (Recursively Enumerable Language) CSL (Context Sensitive Language) CFL (Context Free Language) RL (Regular Language)
Examples of Formal Language simple matching language : Lm = {anbn | n ≥ 0} CFL
double matching language : Ldm = {anbncn | n ≥ 1} CSL
mirror image language : Lmi = {ωωR | ω V∈ T*} CFL
palindrome language : Lr = {ω | ω = ωR } CFL
parenthesis language : Lp = {ω | ω: balanced parenthesis} CFL
The Chomsky Hierarchy of Languages
unrestricted language
context-sensitive language
context-free language
regular language
Languages & Recognizers
Grammar Language Recognizer
type 0(unrestricted)
type 1(context-sensitive)
type 2(context-free)
type 3(regular)
recursively enumerable set
context-sensitive language
context-free language
regular language
Turing Machine
Linear Bounded Automata
Pushdown Automata
Finite Automata